3.3 – Properties of Functions

1. 3.3 – Properties of Functions Precal

2. Review increasing and decreasing: • Increasing function – up when going right • Decreasing function – down when going right • Constant – neither increasing nor decreasing (horizontal)

3. Determine the parts of the graph where the function is increasing, decreasing, and/or constant • Increasing: • Decreasing: • Constant:

4. Local Extrema • Extrema is the plural of extreme • This refers to where the graph reaches peaks and valleys • We call the “peaks” local maximums • We call the “valleys” local minimums

5. What is the local maximum of this function? • Point A is a local maximum because the graph changes from increasing to decreasing at that point • It is only a LOCAL maximum instead of a global maximum because there are points on the graph higher (like point D)

6. What is the local minimum of this function? • Point C is a local minimum because the graph changes from decreasing to increasing at that point • It is only a LOCAL minimum instead of a global minimum because there are points on the graph lower (like point F)

7. Identify the local extrema of the graph • Local Minimums: • C, F, H • Local Maximums: • A, D, G

8. Using the calculator for max’s and min’s

9. Partner Activity • In a little bit you will follow these instructions: • Find a partner • One partner come up and grab a marker • Both partners find a spot at the board • Be prepared to graph some functions

10. Partner Roles • The partner who got the marker is the “player” • The partner without the marker is the “coach” • When I give you the first problem, the coach is going to tell the player how to graph it • Players cannot draw anything unless the coachtells them to do so • Coaches cannot have the marker and draw

11. The “Big Ten” • You are going to graph the ten most important base graphs of functions to remember • This is a part of section 3.4 (I have a handout for you on these graphs that you can use as notes)

12. Functions to graph (1) • Graph f(x) = 1 • Is there any symmetry to this graph? • Can you reflect it over anything?

13. Functions to graph (2) • Graph f(x) = x • Is there any symmetry to this graph? • Can you reflect it over anything?

14. Switch roles • Give the marker to the other partner • The original “player” is now the “coach and vice versa

15. Functions to graph (3) • Graph f(x) = x2 • Is there any symmetry to this graph? • Can you reflect it over anything?

16. Functions to graph (4) • Graph f(x) = x3 • If the coach needs the help of a calculator, that is okay • Is there any symmetry to this graph? • Can you reflect it over anything?

17. Do you notice the pattern of symmetry? • A function with an odd power reflects over the origin • A function with an even power reflects over the y-axis • Go write the red part of this slide in your notes for 3.3, then go back to the board • Switch player-coach roles again

18. Functions to graph (5) • Graph • If the coach needs the help of a calculator, that is okay

19. Functions to graph (6) • Graph • If the coach needs the help of a calculator, that is okay

20. Switch roles

21. Functions to graph (7) • Graph • If the coach needs the help of a calculator, that is okay

22. Functions to graph (8) • Graph • If the coach needs the help of a calculator, that is okay

23. Switch roles

24. Functions to graph (9) • Graph • If the coach needs the help of a calculator, that is okay

25. Functions to graph (10) • Graph • If the coach needs the help of a calculator, that is okay • This is the last one, so return the marker and head back to your seats when you are finished

26. Is this function odd, even, or neither? • Even (reflects over the y-axis)

27. Is this function odd, even, or neither? • Neither – it is not a function, even though it reflects over the x-axis

28. Is this function odd, even, or neither? • Odd – it reflects over the origin